Numerične metode za integracijo hitro oscilirajočih funkcij

delo diplomskega seminarja

Aljoša Rebolj (Author), Marjetka Krajnc (Mentor)

Abstract

V delu diplomskega seminarja si pogledamo numerične metode za integracijo hitro oscilirajočih funkcij specifičnega tipa. Podrobneje obravnavamo dva razreda metod, asimptotske metode in Filonove metode. V obeh primerih integrand ločimo na del, ki povzroča hitro oscilacijo, in del, ki ne oscilira oziroma oscilira počasi. Asimptotske metode so primerne predvsem pri zelo hitrih oscilacijah in se ločijo glede na lastnosti hitro oscilirajočega dela. Ideja Filonovih metod je, da pohleven del integranda aproksimiramo s polinomi, iz preostanka pa izpeljemo tako imenovane momente, ki se jih pod določenimi pogoji da eksaktno izračunati. Vse obravnavane metode implementiramo in preizkusimo na več numeričnih primerih, pri čemer primerjamo absolutno napako glede na različne hitrosti oscilacije.

Keywords

matematika;hitro oscilirajoče funkcije;asimptotska metoda;Filonove metode;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[A. Rebolj]
UDC:	519.6
COBISS:	75600899
Views:	682
Downloads:	64
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Numerical methods for highly oscillatory integrals
Secondary abstract:	In this thesis, we look at numerical methods for the integration of highly oscillatory functions of a specific type. We discuss two classes of methods in detail, asymptotic methods and Filon Methods. In both cases, we divide the integrand into a part, which causes high oscillation, and a part that does not oscillate or oscillates slowly. Asymptotic methods are particularly suitable for very high oscillations and differ according to properties of the part, which causes high oscillation. The idea of Filon methods is that we approximate the part that does not contribute to high oscillation with polynomials, and from the rest of the integrand we derive the so-called moments, which can be calculated analitically under certain conditions. We implement all the discussed methods and test them on several numerical cases, comparing the absolute error with respect to different oscillation speeds.
Secondary keywords:	mathematics;highly oscillatory functions;asymptotic method;Filon methods;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	25 str.
ID:	13335675